The equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(-a(n)), where M and N are integers. Moreover, it is solvable in numbers of the form N + M*sqrt(-l), where l>0 has the form l = A007913(4*k*m^3 - k^4), where k,m >= 1 (without restrictions k,m <= 5*l). But in this more general case there could be unknown numbers l having this form; this circumstance does not allow construction of the full sequence of such l. Therefore we restrict ourselves by the condition k,m <= 5*l. Note that testing l with respect to this condition is rather simple by sorting all values of k,m <= l. One can prove that, at least, in case the Fermat numbers (A000215) are squarefree, the sequence is infinite. Conjecture (necessity of the form of l): If the equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(-l) with integer N,M, then there exist positive integers k,m such that l = A007913(4*k*m^3 - k^4).